Categorical Büchi and Parity Conditions via Alternating Fixed Points of Functors
Categorical studies of recursive data structures and their associated reasoning principles have mostly focused on two extremes: initial algebras and induction, and final coalgebras and coinduction. In this paper we study their in-betweens. We formalize notions of alternating fixed points of functors using constructions that are similar to that of free monads. We find their use in categorical modeling of accepting run trees under the Büchi and parity acceptance condition. This modeling abstracts away from states of an automaton; it can thus be thought of as the “behaviors” of systems with the Büchi or parity conditions, in a way that follows the tradition of coalgebraic modeling of system behaviors.
We thank Kenta Cho, Shin’ya Katsumata and the anonymous referees for useful comments. The authors are supported by JST ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), and JSPS KAKENHI Grant Numbers 15KT0012 & 15K11984. Natsuki Urabe is supported by JSPS KAKENHI Grant Number 16J08157.
- 9.Ghani, N., Hancock, P., Pattinson, D.: Representations of stream processors using nested fixed points. Log. Methods Comput. Sci. 5(3) (2009)Google Scholar
- 11.Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace semantics via coinduction. Log. Methods Comput. Sci. 3(4) (2007)Google Scholar
- 18.Urabe, N., Hasuo, I.: Coalgebraic infinite traces and Kleisli simulations. CoRR abs/1505.06819 (2015). http://arxiv.org/abs/1505.06819
- 19.Urabe, N., Hasuo, I.: Fair simulation for nondeterministic and probabilistic Buechi automata: a coalgebraic perspective. LMCS 13(3) (2017)Google Scholar
- 20.Urabe, N., Hasuo, I.: Categorical Büchi and parity conditions via alternating fixed points of functors. arXiv preprint (2018)Google Scholar
- 21.Urabe, N., Shimizu, S., Hasuo, I.: Coalgebraic trace semantics for Büchi and parity automata. In: Proceedings of the CONCUR 2016. LIPIcs, vol. 59, pp. 24:1–24:15. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)Google Scholar